Spectral properties of second-order, multi-point, p-Laplacian boundary value problems

We consider the multi-point boundary value problem - f{symbol}_p (u^')^' = ? f{symbol}_p (u), on (- 1, 1),u (± 1) = underover(?, i = 1, m^±) a_i^± u (?_i^±), where p > 1, f{symbol}_p (s) {colon equals} | s |^{p - 1} sgn s for s ? R, ? ? R, m^± = 1 are integers, ?_i^± ? (- 1, 1), 1 = i = m^±, and the coefficients a_i^± satisfy underover(?, i = 1, m^±) | a_i^± | < 1 . A number ? ? R is said to be an eigenvalue of the above problem if there exists a non-trivial solution u. The spectrum is the set of eigenvalues. In this paper we obtain some basic spectral and degree-theoretic properties of this eigenvalue problem. These results have numerous applications to more general problems. As an example, a Rabinowitz-type, global bifurcation theorem is briefly described. © 2010 Elsevier Ltd. All rights reserved.